ECE U322 Digital Logic Design

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ECE U322Digital Logic Design
Sept. 26, 2005

• Lecture 9
• Minimization
• Karnaugh Maps
• Reading Marcovitz 3.1

lect09.ppt
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Two Level Logic Minimization

• Goal
• minimize logic in SOP form or POS form
• SOP and POS forms
• same delay from any input to output
• Minimized SOP form
• Fewer Product terms
• fewer AND gates, fewer inputs to OR gate
• Fewer literals in each product term
• fewer inputs to each AND gate

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Karnaugh Map (K-map)

• Diagram made up of squares
• Each square represents a minterm (or maxterm)
• Presents a visual diagram of all possible ways a
  function may be expressed in standard form
• Simplified expressions produced are in SOP or POS
  form
• The simplest expression is one with a minimum
  number of terms and with the fewest possible
  number of literals in each term

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3 Variable K-map

• f(X,Y,Z) m0 m3 m5

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4-Variable Map

• 5-variable and higher K-maps are difficult to
  deal with.

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Simplification with K-Maps

• Find minimal SOP expression
• Equivalent to original expression with
• Minimum number of product terms
• Each product term has minimum number of literals
• We will use K-maps to aid in simplification

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Adjacency and Kmaps

• Definition
• Two minterms are logically adjacent if they
  differ in one variable position
• Property
• Two logically adjacent terms can be combined to
  eliminate one variable
• In K-Maps, logically adjacent variables are
  physically adjacent

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Guidelines for simplifying K-Maps

• Each square on a K-Map of n variables has n
  squares which are logically adjacent
• Explore rectangles (collection of adjacent
  squares) in powers of 2 (such as 1, 2, 4, 8, )
• A rectangle of
• 2 squares eliminates ___ variables
• 4 squares eliminates ___ variables
• 2n squares eliminates ___ variables
• A rectangle encompassing the entire map produces
  a function that is always equal to logic ___

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Guidelines (contd)

• Group as many squares as possible.
• A larger group
• Make as few groups (or rectangles) as possible to
  cover all the minterms.
• Fewer groups
• Minterm is covered if it is included in at least
  one rectangle.
• Begin with
• squares with the fewest adjacencies or
• largest groups

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• Example
• Simplify F(X,Y,Z) Sm(0,2,4,5,6).

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• Example
• Simplify F(X,Y,Z) Sm(1,2,5,7)

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• Example
• Simplify F XZ XY XYZ YZ

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• Sometimes, there are alternative ways to combine
  squares.

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• Example Simplify
• F(W,X,Y,Z) Sm(0,1,2,4,5,6,8,9,12,13,14).

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Map Manipulation

• Implicant
• is a product term used to cover minterms.
• Rectangles made up of squares containing 1s
  correspond to implicants.
• Prime Implicant
• implicant not covered by any other implicant.
• Essential Prime Implicant
• covers at least one minterm not covered by any
  other prime implicant.

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Goal Find a cover

• Cover
• set of prime implicants that covers each minterm
  at least once.
• Note some minterms can be covered more than
  once.
• Find a cover
• all minterms covered at least once (possibly
  more)
• all essential prime implicants must be included
• fewest groups that cover all the minterms

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F(X,Y,Z) S m(0,2,4,6,7)
1
1
1
1
1
Implicants Prime Implicants Essential Prime
Implicants Cover
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Systematic procedure for simplifying K-Maps

• Determine all prime implicants.
• Identify and select all essential prime
  implicants.
• Select and minimize subset of remaining
  nonessential prime implicants to cover those
  minterms not covered by EPI.

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Nonessential prime implicants

• Selection Rule
• Minimize the overlap among prime implicants.
• Make sure each prime implicant selected includes
  at least one minterm not included in any other
  prime implicant selected.

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F(X,Y,Z) S m(1,3,4,5)
1
1
1
1
Implicants Prime Implicants Essential Prime
Implicants Cover
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• Example Simplify
• F(A,B,C,D) Sm(0,2,5,7,8,10,13,15).

C
CD
AB
B
A
D
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• Example Simplify
• F(A,B,C,D) Sm(0,1,2,4,5,10,11,13,15).

C
CD
AB
B
A
D
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• Example Simplify
• F(A,B,C,D) Sm(0,2,3,8,10,11)

C
CD
AB
B
A
D
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• Example Simplify
• F(A,B,C,D) Sm(0,1,2,5,10,11,13,15)

C
CD
AB
B
A
D
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Product-of-Sums Simplification

• Similar to SOP simplification
• Instead of finding groups of 1s, find groups
  of 0s
• Finding POS form
• If input variable is 0 -gt not complemented
• If input variable is 1 -gt is complemented
• Thes are called implicates (similar to
  implicants)

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• Example
• Map F (A B C)(B D).

C
CD
AB
B
A
D
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Map Manipulation (POS form)

• Implicate
• is a sum term used to cover Maxterms.
• Rectangles made up of squares containing 0s
  correspond to implicates.
• Prime Implicate
• implicate not covered by any other implicate
• Essential Prime Implicate
• covers at least one Maxterm not covered by any
  other prime implicate

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Goal Find a cover (POS)

• Cover
• set of prime implicates that covers each Maxterm
  at least once.
• Note some Maxterms can be covered more than
  once.
• Find a cover
• all Maxterms covered at least once (possibly
  more)
• all essential prime implicates must be included
• fewest groups that cover all the Maxterms

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• Example Simplify the following in POS form
• F(A,B,C,D) ?M(0,1,2,5,8,9,10).

C
CD
AB
B
A
D
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• Example Simplify the following in POS form
• F(A,B,C,D) ?M(0,1,2,3,6,9,14).

C
CD
AB
B
A
D
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• Example Simplify
• F(W,X,Y,Z) PM(0,1,3,5,7,9,12).

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Dont-Care on Outputs

• Incompletely specified functions
• functions that have unspecified outputs for some
  input combinations.
• Occurs when
• some input combinations never occur.
• we do not care what the output is for some input
  combinations that are expected to occur.
• Dont-Care conditions
• the unspecified minterms of a function.
• symbolized by an X.

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Dont cares on outputs

• Example BCD to 7-segment decoder
• In lab, hex to 7-segment decoder
• all input combinations have specific output
  values that are required
• BCD to 7-segment decoder
• only display digits 0 through 9
• input combinations A through F should never occur

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BCD to 7-segment decoder

• Input combinations A through F should never occur
• How to deal with them?
• Always set outputs to zero (done in book)
• Set outputs that correspond to input combinations
  A through F
• minterms _____________________to dont care
  outputs

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Dont cares on outputs

• Dont cares
• As a designer, I dont care if the output is a 1
  or a 0, because it should never arise
• I will choose the output to either be a 1 or
  0 whichever way simplifies my circuit
• Example Segment e on BCD to seven segment
  decoder

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• Last time BCD-to-Seven-Segment Decoder,
  Segment e ?m( )

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• Example Simplify Segment e
• F(A,B,C,D) Sm( ).

C
CD
AB
B
A
D
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• Now simplify Segment e using A-F as dont cares
  F(A,B,C,D) Sm( ) d(10 -15)

C
CD
AB
B
A
D
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• Example Simplify F(A,B,C,D) Sm(1,3,7,11,15)
  d(0,2,5)

C
CD
AB
B
A
D
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• Example Simplify F(A,B,C,D) ?M(0,2,6,8,10)
  d(12,13)

C
CD
AB
B
A
D